Evaluating $\int_0^\infty \frac{\arctan (x^2) \ln(1+x^4)}{x(x^8+x^4+1)} \mathrm dx$

Question

So, my friend has challenged me to solve the following integral

$\displaystyle \tag*{} I=\int_0^\infty \frac{\arctan (x^2) \ln(1+x^4)}{x(x^8+x^4+1)} \mathrm dx$

I started by doing $x^2=t$, so

$\displaystyle \tag{1} I=\frac 12 \int_0^\infty \frac{\arctan (x) \ln(1+x^2)}{x(x^4+x^2+1)} \mathrm dx$

Now, I first tried to solve this generalized integral:

$\displaystyle \tag*{} \mathcal L(m,n,a) = \int_0^\infty \frac{\arctan(mx)\ln(1+n^2x^2)}{x(a^2+x^2)} \ \mathrm dx$

We have $\mathcal L(0,1,a)=0$ and $\mathcal L(1,0,a)=0$. We now differentiate $\mathcal L(m,n,a)$ w.r.t $m$ first and then w.r.t $n$. We get:

$\displaystyle \tag{2} \begin{align} \partial m \ \partial n \ \mathcal L(m,n,a) &= \int_0^\infty \frac{2nx^2}{(a^2+x^2)(1+m^2x^2)(1+n^2x^2)} \ \mathrm dx \\\\ &= \frac{\pi n}{(m+n)(am+1)(an+1)} \\\\ &= \frac{\pi}{2(am+1)(an+1)} , \ \ \text{via symmetry} \end{align}$

And now taking the double integeral gives $\displaystyle \tag{3} \mathcal L(1,1,a) = \int_0^1\int_0^1 \frac{\pi}{2(am+1)(an+1)} \ \mathrm dm \ \mathrm dn$ Using the elementary result: $\displaystyle \tag*{} \int_0^1 \frac{1}{ay+1} \ \mathrm dy = \frac{\log(a+1)}{a}$ We find the value of $(3)$ as $\displaystyle \tag*{} \mathcal L(1,1,a) = \frac{\pi \log^2(a+1)}{2a^2}$ Finally, we conclude that $\displaystyle \tag*{}\int_0^\infty \frac{\arctan(x)\ln(1+x^2)}{x(a^2+x^2)} \ \mathrm dx={\color{Red} { \frac{\pi \log^2(a+1)}{2a^2}}}:= {\color{Blue}{ f(a)}}$

Now by completing square and performing, we have:

$\displaystyle \tag*{} \begin{align}\frac{1}{x^4+x^2+1} &=\frac{1}{\left(x^2+\frac 12\right)^2+\frac 34 }\\ &= \frac{1}{\left(x^2+\frac 12 - \frac{i \sqrt 3}{2}\right)\left(x^2+\frac 12 + \frac{i \sqrt 3}{2}\right)} \\ &= \frac{2}{i \sqrt 3} \left(\frac{1}{\left(x^2+\frac 12 - \frac{i \sqrt 3}{2}\right)}-\frac{1}{\left(x^2+\frac 12 + \frac{i \sqrt 3}{2}\right)}\right)\end{align} $

Adding all the pieces together, we get

$\displaystyle \tag{4} I=\frac{1}{i \sqrt 3} \left(f\left(\sqrt{\frac{1-i\sqrt3}{2}}\right) - f\left(\sqrt{\frac{1+i\sqrt3}{2}}\right)\right)$

My question is to simplify $(4)$ (if all my steps are correct) and also is there any short ways (methods others than contour integration) to destroy this integral?

Thanks :).

Answer 1

Note that $a_\pm =\sqrt{\frac{1\pm i\sqrt3}{2}}=\frac{\sqrt3\pm i}2$ and \begin{align} I=&\ \frac 12 \int_0^\infty \frac{\arctan x\ln(1+x^2)}{x(x^4+x^2+1)} dx\\ =& \ \frac{1}{2\sqrt 3 i} \int_0^\infty \frac{\arctan x\ln(1+x^2)}{x(x^2+a_-^2)}-\frac{\arctan x\ln(1+x^2)}{x(x^2+a_+^2)}\ dx\\ = & \ \frac\pi{4\sqrt3 i}\bigg(\frac{\ln^2(a_- +1)}{a_-^2}- \frac{\ln^2(a_+ +1)}{a_+^2} \bigg)\\ =& \ \frac\pi{4\sqrt3 i}\bigg(\frac{\ln(a_- +1)}{a_-}- \frac{\ln(a_++1)}{a_+} \bigg)\bigg(\frac{\ln(a_- +1)}{a_-}+\frac{\ln(a_+ +1)}{a_+} \bigg)\\ =& \ \frac\pi{4\sqrt3 i}\cdot\frac{\sqrt3 \pi -6\ln(2+\sqrt3)}{12i} \cdot \frac{6\sqrt3\ln(2+\sqrt3)+\pi}{12}\\ = &\ \frac\pi{16}\ln^2(2+\sqrt3)-\frac{\pi^2}{48\sqrt3}\ln(2+\sqrt3)-\frac{\pi^3}{576} \end{align}

Answer 2

Note that

$$\Im\left(-\frac{2}{\sqrt{3}\left(x^2+\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)} \right) = \frac{1}{x^4+x^2+1}$$

Hence

$$ I=\frac 12 \int_0^\infty \frac{\arctan (x) \ln(1+x^2)}{x(x^4+x^2+1)} \mathrm dx = \Im \left( -\frac{1}{\sqrt{3}} \int_0^\infty \frac{\arctan (x) \ln(1+x^2)}{x\left(x^2+\frac{1}{2}+\frac{i\sqrt{3}}{2}\right)} \mathrm dx \right) $$

which is just the imaginary part of

$$ -\frac{1}{\sqrt{3}}\int_0^\infty \frac{\arctan(x)\ln(1+x^2)}{x(a^2+x^2)} \ \mathrm dx={\color{Red} { -\frac{\pi \log^2(a+1)}{2\sqrt{3}a^2}}}$$

with $ \displaystyle a = \sqrt{\frac{1}{2}+\frac{i\sqrt{3}}{2}}$

Hence

\begin{align*} I =& \Im\left(-\frac{\pi \left(\frac{1}{2}-\frac{i\sqrt{3}}{2} \right)\log^2\left(1 + \sqrt{\frac{1}{2}+\frac{i\sqrt{3}}{2}}\right)}{2\sqrt{3}}\right) \\ =& \Im\left(-\frac{\pi \left(\frac{1}{2}-\frac{i\sqrt{3}}{2} \right)\left[\ln\left|1 + \sqrt{\frac{1}{2}+\frac{i\sqrt{3}}{2}}\right| + i\arg\left(1 + \sqrt{\frac{1}{2}+\frac{i\sqrt{3}}{2}}\right) \right]^2}{2\sqrt{3}}\right)\\ =& \Im\left(-\frac{\pi \left(\frac{1}{2}-\frac{i\sqrt{3}}{2} \right)\left[\frac{1}{2}\ln(2+\sqrt{3}) + i\frac{\pi}{12}\right]^2}{2\sqrt{3}}\right)\\ =& \Im\left(-\frac{\pi \left(\frac{1}{2}-\frac{i\sqrt{3}}{2} \right)\left[\frac{\log^2(2+\sqrt{3})}{4} + \frac{ \pi i\ln(2+\sqrt{3})}{12} - \frac{\pi^2}{144} \right]}{2\sqrt{3}}\right)\\ =& -\frac{\pi^2\ln(2+\sqrt{3})}{48\sqrt{3}}+ \frac{\pi\ln^2(2+\sqrt{3})}{16} -\frac{\pi^3}{576} \end{align*}